Exploring Amputated Connected Graphs in QFT

[PRB147]

In Peskin and Schroder Book QFT:
They used the term: "amputated connected graphs".
Does amputated connected graph in Peskins's book
is same as the irreducible graph in common sense?
I think it is the same, please reply. thank you all!

 

[Simon_Tyler]

I'm not sure what you mean by "irreducible graph in common sense".

In QFT you often talk about one-particle-irreducible graphs which are closely related to the quantum effective action. These are graphs that can not be disconnected by simply cutting a single line (propagator). Connected graphs (eg tree graphs) can be disconnected by cutting a single line. Amputated means that you remove the external lines but keep the momenta that flowed through them -- it is not related to connectedness.

 

[Bill_K]

Quoting from another text,

Consider any diagram M. A part of M which is connected to the rest of M by exactly two lines is called an inserted SE-part. Similarly, a part of M which is connected to the rest of M by exactly two electron lines and one photon line is called an inserted V-part. To every M there corresponds a uniquely defined diagram M' called the skeleton of M, which is obtained from M by replacing every inserted SE-part by a line, and every inserted V-part by a corner...

A diagram or diagram part is called irreducible if it is equal to its own skeleton. It is called reducible is this is not the case. A reducible SE- or V-part can be either proper or improper. It is called improper if it can be separated into two disconnected parts by the omission of one single line. Diagrams for which this is not the case are called proper.

 

[Simon_Tyler]

Too many words with conflicting definitions. Too many physicists making up terminology.

Dy the way, which text is that? I don't think I've ever heard that terminology before.

I remember skeleton of a graph to mean the tree graph obtained obtained from a general graph by contracting every one-particle irreducible (1PI) subgraph to a single point.
The reverse of this (inserting 1PI graphs into a tree) is how you use 1PI graphs obtained from the effective action to construct arbitrary amplitude. This shows how the effective action contains all of the information that is in the full generating functional.

 

[PRB147]

Thank you all for the replies.
my "irreducible" here means the proper.
see Peskin's Book, amputated graphs do have external lines, see the graph
in page 114 of their book.

 

[Simon_Tyler]

@PRB147:

So irreducible means 1-particle irreducible. (Or in graph theory speak: 2-connected).

As for amputated graphs and external lines, it depends what you mean be external lines.
The diagram in P&S shows that you truncate all parts of the graph that make up contributions to external full propagators. This includes the "inner-most external line" which has to be a perturbative propagator. All that's left is the momentum that would have been running through the external propagators and the vertex that you're interested in. This external momenta does not really correspond a full external edge, since an edge --> propagator.

I've seen more mathematical texts talk about how the Feynman diagram way of creating graphs is useful. You start off with half-edges (fields) which are then joined to form full edges (Wick contract two fields to get a propagator). In this sense, the external lines are half-edges -- ie places where you can attach another half-edge in order to get the expression needed for scattering amplitudes.

This point of view works even better when thinking of the quantum effective action. Then the external lines correspond to the classical/external/background field - i.e. the arguments of the effective action.

 

[Bill_K]

Too many words with conflicting definitions. Too many physicists making up terminology. Dy the way, which text is that? I don't think I've ever heard that terminology before.

Now you have. I believe the physicist in question is Freeman Dyson.

I remember skeleton of a graph to mean the tree graph obtained obtained from a general graph by contracting every one-particle irreducible (1PI) subgraph to a single point

You got it partly right, but in general the skeleton is not a tree. Which is what gives rise to overlapping divergences.

 

[elliot3019]

I have a somewhat related question. If anyone is familiar with Srednicki's textbook (which takes the path integral approach), I am wondering where he accounts for need to amputate the diagrams, or what corresponding step is made. In Peskin and Schroeder the amputation is related to the LSZ formula, but Srednicki never mentions amputation in relation to the LSZ formula or Feynman diagrams (sections 5, 9-10 of Srednicki).